fourier analysis- summary

[Rosemary Reid]

Dear Biomch-l,

Thank you to everyone who replied to my query about fourier analysis.
I think that by using cos and sine theta instead of cos and sine t I was
confusing everything so I am now using data which includes t.

This is a summary of the replies:

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Those who replied:



Christian Peham
University of Veterinary Medicine Vienna

Giannis Giakas PhD
Sport Health and Exercise
School of Health
Staffordshire University

At Hof
Department of Medical Physiology
University of Groningen


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Book suggestions


When you are
interested in programming a FFT, sou should have a look at the following
book: W.H.Press: "Numerical Recipies", Cambridge Press.

Hope this helps you. Regards

Martin

--
__________________________________________________ _____

Martin Pohl
Abteilung Unfallchirurgische Forschung und Biomechanik
Universitaet Ulm
Hemholtzstrasse 14
D-89081 ULM
Tel.: +49-(0)731-502-3494
FAX: +49-(0)731-502-3498
WWW: http://lyra.medizin.uni-ulm.de
e-mail: mailto://pohl@pollux.medizin.uni-ulm.de
__________________________________________________ _____




At first blush I would say there is nothing wrong with what you are trying
to do. One can apply the FT to any sort of periodic data. One question
I have is how to treat repeated measurements. Suppose you make measurements
both as the leg swings forward and backwards. Presumably these data would
be analysed separately. Would one average the data for the leg swinging
forward from several walking cyles? This seems to be appropriate since
you are trying to derive a time independent analysis of F(theta).

Peter HardyI would say that you can go on doing Fourier Analysis to the the
theta angle as the time axis. The same concept is employed e.g. for
spatial differentiation and spatial z-transformation, where time is
substituted by distance.

The only thing you should bear in mind is the new meaning of your
frequency. It is not a cyclic oscilation (sine/cosine) with variable
cycle duration. Instead, it will be some kind of angular frequency
difficult for me to conceptualize.

Ruben Lafuente-Jorge
Institute of Biomechanics of Valencia
P.O. Box 199
46980-Paterna (Spain)
E-Mail: rlafuent@ibv.upv.es
Tel: (96) 1366032
Fax: (96) 1366033
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A very useful program in Matlab

hi,
as i understand fourier analysis -- the x axis can be anything
(time, angle, 5 gait cycle). i recently carried out a similar analysis on
gait data. i wrote several matlab functions -- i have attached one that
calculates the fourier coeff of a signal, if u're using matlab ie.

kavi bhalla
----------------------------------------------------------------------------
Kanwaljeet Singh Bawa Bhalla | Home Address :
Phd Student (biomechanics) | # G4, 301 Maple Ave
Mechanical and Aerospace Engineering | Ithaca, NY 14850
Cornell University | Ph 607)256-7109
****************** email: ksb10@cornell.edu *****************************
************* http://talus.mae.cornell.edu/~bhalla ***********************
-----------------------------------------------------------------------------
"koi kisi ko raj na de hai, jise miley hai nij bal se miley hai"
Content-Type: TEXT/PLAIN; charset=US-ASCII; name="fourier.m"

Content-ID:
Content-Description:

%fourier.m
% when given an arbitrary signal , generates the fourier coeff for the
% signal. Based on "Vu and Esfandiari: Dynamic Systems : Modelling and
% Analysis, McGraw Hill pg 385."
%
% INPUTS : data : Nx2 matrix : signal : col1 : time
% col2:signal
% N : No of fourier coefficients to be used.
% typcially between 1000 and 5000
% T : Time Period. end time for most signals
% OUTPUTS :ao : const
% a : Nx1 matrix
% b : Nx1 matrix
function [a0,a,b]=fourier(data,T,N)
[n,junk]=size(data(:,1));
endtime=data(n,1);

ti=linspace(0,endtime,N);
f1=interp1(data(:,1),data(:,2),ti);

p=T;
a0=0;
a=zeros(N,1);
b=zeros(N,1);
deltat=T/N;
a0= a0+ sum(f1)*deltat*2/p;

for i=1:N
a(i)=a(i)+ sum(f1.*cos(2*i*pi*ti'/p))*deltat*2/p;
b(i)=b(i)+ sum(f1.*sin(2*i*pi*ti'/p))*deltat*2/p;
end;

% the following (if activated) generates a sample based on the above
% generated fourier coeffs and plots them.
%
% f=ones(N,1)*a0/2;
% for i=1:N
% f=f+ a(i)*cos(2*i*pi*ti'/p) + b(i)*sin(2*i*pi*ti'/p);
% end
% hold on
% plot(ti,f1,'xr')
% plot(ti,f) % generated



Dear Rosemary
I have just seen your message because I was away so here are my late
comments:
The equation you describe is correct if your data series is defined
over 0-360 deg. By entering N data points a standard FFT algorithm
will calculate N/2 harmonics. The problem is that if you do not
specify the period of your data in seconds you cannot calculate the
frequency of those harmonics in Hz: Fn=(1/T)*n where n is the
frequency number (n=1..N/2), T is your period and Fn is the frequency
in Hz of the nth harmonic.
Expressed relative to time the equations are:
F(t)=Ao+Sum(Ansin(njt)+Bncos(njt))
where n=1..N/2 and j is the angular frequency (j=360/T or j=2pi/T
if you are working in radians)
Matlab or any other numerical analysis tool would do the job
although you have to be aware that some FFT algorithms are working
for number of data points equal to a power of two only. If you have
any problems finding a suitable algorithm Email your data to me and I
will run an appropriate Fourier Transformation.
Finally an excellent reference on FT is a book called: "The FFT
Fundamentals and Concepts" edited by R. Ramirez, Prentice-Hall.
All the best

Bill

Dr V. Baltzopoulos
Reader in Biomechanics
Department of Exercise and Sport Science
Manchester Metropolitan University
Hassall Rd
Alsager ST7 2HL
Great Britain

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Rosemary Reid
Department of Mechanical Engineering
Loughborough University
Loughborough
Leicestershire
LE11 3TU
U.K.

tel. 44 1509 223748